(1) Recap

• correlation
• two problems:
  • confounding/bias
  • random association

(2) Solutions for Bias

• design-based vs. adjustment-based
• conditioning
  • what is it?
  • how does it work?

1. We start with a causal claim

2. Turn claim into a causal theory
   • causal logic, independent \(X\)/dependent \(Y\) variables
3. Turn causal logic and \(X\)/\(Y\) into hypotheses
   • expect that potential outcomes of \(Y\) change with \(X\)

4. B/C of FPCI: imperfectly test hypotheses using correlation of observed values of \(X\) and \(Y\)

5. Infer causality if:
   • assumptions about cases we compare let us ignore confounding/bias
   • correlation unlikely to have occured by chance

Bias/Confounding

Equivalently:

1. Observed cases with different values of \(X\) have different potential outcomes of \(Y\)
2. Observed cases with different values of \(X\) have different values of \(W\), which is related to \(Y\)
3. There is a "backdoor" causal path from \(X\) to \(Y\)

Random Error

1. Correlation between \(X\) and \(Y\) is by chance and reflects no change in \(Y\) due to \(X\).

Bias/Confounding

1. adjustment-based solutions
2. design-based solutions

Random Error

1. Increase number of independent cases
2. Take courses in statistics

adjustment-based

• Identify possible confounding variables (e.g. \(W, Z, V, U\))
• Measure these variables
• adjust correlation of \(X\) and \(Y\) by "conditioning" on confounding variables

design-based

• Compare cases that, by assumption, are
  • similar in terms of confounding variables \(W\)/ potential outcomes of \(Y\)
  • exposed to \(X\) in a manner unrelated to \(W\)/potential outcomes of \(Y\)

Adjustment-based approaches start from the…

Comparative Method:

Sometimes called the "method of difference" (via John Stuart Mill), this assesses whether \(X\) causes \(Y\) by…

• comparing two cases that are the same in all relevant respects, except for value of \(X\)
• assess for these two cases whether \(Y\) changes when \(X\) changes (correlation)

Question:

What causes the spread of cholera?

Causal claim:

Contaminated water causes cholera outbreaks

Obvious?

19th Century London saw repeated outbreaks of cholera, with mass death

• Dominant view was that "miasmas" or bad air caused diseases like cholera

{.centered}

John Snow, MD

Snow could compare places with contaminated water with those that did not, but there is a possibility of confounding:

• rotting organic material \(\to\) miasmas \(\to\) cholera?
• rotting organic material \(\to\) contaminated water \(\to\) cholera?

Comparing observed water quality and cholera in:

• two different areas with different water quality
• same area at different times with different water quality

would not rule out miasma as explanation for cholera.

• "miasma" changes as "water" changes; might affect cholera; \(\to\) confounding

Near the outbreak

Far from outbreak

A more general approach to the comparative method is:

conditioning

when we observe \(X\) and \(Y\) for multiple cases, we examine the correlation of \(X\) and \(Y\) within groups of cases that have the same values of confounding variables \(W, Z, \ldots\).

How does conditioning solve the problem? Equivalently:

• Cases compared have same potential outcomes of \(Y\) for different values of \(X\)
• Cases compared have SAME values on confounding variables \(W\); \(W\) cannot be related to \(X\) or to \(Y\) \(\to\) no confounding
• "Backdoor" path from \(X\) to \(Y\) is "blocked"

Sometimes we think about "conditioning" like this:

• conditioning lets us find the correlation between \(X\) and \(Y\), ceteris parabis, "all else being equal".

But all else does not need to be equal:

• We only need to compare cases that are the same on confounding variables
• same on variables that are causally linked to \(X\) and \(Y\)

Sanctuary Cities

• Trump Administration intensified immigration enforcement
  
• Some Cities/Counties opted to not assist US Immigrations and Customs Enforcement (ICE) by detaining immigrants for longer
• These are "Sanctuary Cities"

Effect on Crime

• Trump and political allies suggest "sanctuary" policies undermine law enforcement, lead to more crime

Does having "sanctuary" policy increase crime?

If we simple look at the unadjusted difference in crime in sanctuary vs. non-sanctuary counties, we would get:

\[\frac{\overbrace{35 + 35 + 5}^{\text{Mean Crime in Sanctuary}}}{3} - \frac{\overbrace{40 + 10 + 10}^{\text{Mean Crime w/out Sanctuary}}}{3} = \frac{15}{3} \neq -5\]

This is bias and must result from confounding

\(Urban_i\) is a confounding variable. Why?

If we condition on \(Urban\) (compare \(X\) and \(Y\) for cases with same value of \(Urban\)), we can remove the bias.

If we condition on \(Urban\), we get the true effect of \(-5\):

\[\frac{\overbrace{35 + 35}^{\text{Sanctuary (Urban)}}}{2} - \frac{\overbrace{40}^{\text{w/out Sanctuary (Urban)}}}{1} = -5\]

\[\frac{\overbrace{5}^{\text{Sanctuary (Not Urban)}}}{1} - \frac{\overbrace{10 + 10}^{\text{w/out Sanctuary (Not Urban)}}}{2} = -5\]

Why does this work?

When we condition on \(W\) (Urban), by looking at cases where \(W\) is the same:

• there can be no relationship between \(W\) and potential outcomes of \(Y\) (Crime)
• there can be no relationship between \(W\) and \(X\) (sanctuary status)

Without these two relationships, confounding does not happen, there is no bias.

What assumptions must we make

in order for adjustment solutions to confounding/bias to work?